Stein's method for comparison of univariate distributions

TL;DR

This paper introduces a unified and general framework for Stein's method to compare univariate distributions, providing a versatile toolkit for distributional approximation and comparison.

Contribution

It proposes a canonical Stein operator based on linear difference or differential operators, unifying continuous and discrete cases, with extensive theoretical and practical applications.

Findings

Developed a new canonical Stein operator for univariate distributions

Provided abstract theorems for distributional approximation

Illustrated the method with comparisons between various distribution pairs

Abstract

We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on a linear difference or differential-type operator}. The resulting Stein identity highlights the unifying theme behind the literature on Stein's method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our approach is illustrated for comparison of several pairs of distributions : normal vs normal, sums of independent Rademacher vs normal, normal vs Student, and maximum of random variables vs exponential, Frechet and Gumbel.